Dual Ramsey, an introduction – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Introduction

Throughout the DocCourse we have primarily focused on Fraïssé limits of finite structures. As we saw in Solecki’s first lecture (not posted yet), it makes sense, and is useful, to consider Fraïssé limits in a broader context. Today we will discuss those other contexts.

Solecki’s first lecture discussed how to take projective Fraïssé limits. Panagiotopolous’ lecture (not posted yet) looked at a specific application of these projective limits. We will see how to take metric (direct) Fraïssé limits.

Overview

Discrete

Compact

Metric Structure

Size

Countable

Separable

Separable, complete

Limit

Fraïssé limit

Quotient of the projective limit

(direct or projective) Metric Fraïssé limit

Homogeneity

(ultra)homogeneity

Projective approximate homogeneity

Approximate homogeneity (*)

Automorphism group

non-archimedian groups (closed subgroups of $S^\infty$

homeomorphism groups

Polish Groups

KPT, extremely amenable iff

RP

Dual Ramsey

Approximate RP (**)

Metrizability of UMF iff

finite Ramsey degree

(***)

(Open) Compact RP?

Where we’ve seen these

Classical

Solecki’s lectures

These lectures

(*) – Exact homogeneity is often not possible.
(**) – In the projective setting this is fairly unexplored. These proofs are usually via direct (discrete) Ramsey, or through concentration of measure.
(***) – You have KPT before you take the quotient, but lose it after taking the quotient. e.g. UMF(pre-pseudo arc) is not metrizable (through RP). A question of Uspenskij asks about the UMF(pseudo arc).

Continuous Logic definitions

In the context of Banach spaces, it makes sense to use continuous logic. This is where we instead of the usual $\{0,1\}$-valued logic we allow sentences to take on values in the interval $[0,1]$. We also suitably adjust the logical constructives.

Classical logic

Continuous logic

True

0

False

1

$=$

$d$

$x \vee y$

$\min\{x,y\}$

$x \wedge y$

$\max\{x,y\}$

$\neg x$

$1-x$

$x \Rightarrow y$

$(x-y) \vee 0$

$\forall$

$\sup$

$\exists$

$\inf$

Now we define functions and relations. Let $(A,d)$ be a complete metric space. So $(A^n, d)$ will be given the sup metric.

$F: A^n \rightarrow A$ comes with a Lipschitz constant.

$R: A^n \rightarrow [0,1]$ comes with a Lipschitz constant.

Then functions and relations must satisfy the usual things that functions and relations satisfy in classical logic.

Examples

Finitely generated substructures

Limit

maps

Language

Separable metric spaces

finite metric spaces

Separable Urysohn space

isometric embedding

just the distance

Separable Banach spaces

finite dimensional Banach spaces (**)

Gurarij space

isometric linear embedding

$\{|| \cdot ||, +, (\cdot \lambda)_{\lambda \in \mathbb{Q}}\}$

Separable Choquet spaces

finite dimensional simplices

Poulsen simplex

affine homeomorphisms (*)

Something that captures the convex structure

(*) – An affine homeomorphism sends $S_0 \rightarrow S_1$ and sends extreme points to extreme points, then is extended affinely to the rest of the simplex. The metric here is not canonical.
(**) – Similar to the discrete case, to take a limit you only need a cofinal sequence. In this case we take $\ell^n_\infty$.

Morphisms between models

In continuous logic the maps between models are isometric embeddings that preserves functions and relations.

Properties of the finitely generated substructures

In the classical Fraïssé setting we looked at homogeneity, HP, JEP and AP. These notions have suitable generalizations in the metric Fraïssé setting.

Definition. Let $(A,d)$ be a metric structure. We describe finitely generated substructures in $(A,d)$ by $\langle \vec{a} \rangle$, where $\vec{a}$ is an $n$-tuple in $A$.

We say that $(A,d)$ is approximately ultrahomogeneous (AUH) if $\forall \vec{a} \in A^n, (\forall n)$ and for every $\phi: \langle \vec{a} \rangle \rightarrow A$ morphism, and for all $\epsilon >0$, there is a $\hat{\phi} \in \text{Aut}(A)$ such that $d(\phi(\vec{a}), \hat{\phi}(\vec{a}))<\epsilon$.

$\text{Age}(A)$ is the collection of finitely generated substructures of $A$.

Lemma. If $A$ is AUH and separable, then $\text{Age}(A)$ has

HP,

JEP,

NAP (the Near Amalgamation Property),

PP (the Polish Property, an analogue of countability).

We now explain NAP and PP. The NAP is a striaghtforward generalization of AP.

Definition. Let $\mathcal{K}$ be a collection of finitely generated metric structures. We say that $\mathcal{K}$ satisfies NAP if when $f_i : A \rightarrow B_i$ are embeddings, then

Definition. Let $\mathcal{K}$ be a collection of finitely generated metric structures with JEP. Define $K_n$ to be all pairs $(\vec{a}, A)$ where $\vec{a} \in A^n$ and $\langle \vec{a}\rangle = A$. Define $d_n$, a pseudometric on $K_n$ by
$$d_n((\vec{a}, A), (\vec{b}, B)) := \inf\{d_c(f(\vec{a}, g(\vec{b}))\},$$
where this is taken over al $C \in \mathcal{K}$ such that $A,B$ embed in $C$, and all embeddings $f: A \rightarrow C$, $g: B \rightarrow C$.

We say $\mathcal{K}$ satisfies the Polish Property (PP) if $(K_n, d_n)$ is separable for all $n$.

The Urysohn space

Recall that $(\mathbb{U}, d)$ is the separable Urysohn space. It is the (unique) complete, separable metric space, universal for separable metric spaces and (exactly) ultrahomogeneous with respect to finite metric spaces.

Its age is the collection of finite metric spaces. It is a metric Fraïssé class.

Its automorphism group has a similar universal property.

Theorem (Uspenskij). $\text{Iso}(\mathbb{U})$ is universal with respect to second countable topological groups.

Definition. Let $\mathcal{K}$ be a collection of finitely generated metric structures. For $A,B \in \mathcal{K}$, $\text{Emb}(A,B)$ is the collection of all morphisms from $A$ to $B$. There is a suitable distance between embeddings which we will not define here (in the special case of Banach spaces it is the operator norm).

Recall that in the infinite case, rigidity was needed to have the embedding RP. That is why in finite metric spaces we added linear orders to get the RP. However, metric spaces do satisfy the ARP (by Pestov from extreme amenabilty of $\text{Iso}(\mathbb{U},d)$, without needing to add linear orders.

Also, by using the usual compactness arguments, we can assume that the witness $C$ to ARP is the full Fraïssé limit.

So we can reword the ARP for finite metric spaces, by transfering the colouring $c: \text{Emb}(A,\mathbb{U}) \rightarrow [r]$ to a colouring $\hat{c} : G / \text{Stab}(A) \rightarrow [r]$.

Thick sets

Thickness is a group property that captures some Ramsey properties. This is desirable because we would like to be able to detect Ramsey type phenomena from the group itself, without having to know the underlying Fraïssé limit.